Oligopoly is a market structure; monopolistic competition is another market structure. They compare in that each is a type of market structure. Both operate in markets with imperfect competition. Both have downward sloping demand curves due to price elasticity and substitution alternatives.
In monopolistic competition many firms within one industry compete against each other with essentially the same product but with each having distinctive characteristics. A well known example is restaurants in the food industry, all competing for diners while offering unique characteristics geared to market segments or to price point.
In contrast with this, in an oligopoly a few large firms dominate an industry, with perhaps a few small firms geared to niche or specialized market segments. These few large firms are identified and recognizable by market share concentration ratios. Gasoline suppliers in the fuel industry are a common example. Large gasoline giants, like the Exxon Mobil gasoline stations, have large market share concentration ratios while independent stations have small market share concentration ratios.
Firms in monopolistic competition have differentiated products, operate independently, and offer no barriers to new entrants. Imagine new entrants to the restaurant industry or to the hair salon industry. In contrast, firms in an oligopoly make minimal attempts to differentiate product (for clarification, think of drive-through restaurants as differentiated from high price-point restaurants); they operate through interdependence and have distinguishing pricing strategies as they decide to compete or collude, to be a first mover or a second mover; they have natural and artificial barriers to new entrants such as economies of large scale (natural) and predatory pricing (artificial).